K-12 Teaching and Learning From the UNC School of Education

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Problem centered math
Why students must build their own understanding of mathematics if they are to be able to use it in the real world, and how teachers can guide them in doing so.
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All meaningful mathematics learning is imaged-based. While there may be certain forms of mathematical reasoning that seem not to use imagery, most mathematical activity has a spatial component. If school mathematics is procedural, students may fail to develop their capacity to form mental images of mathematical patterns and relationships. It is well documented that students who reason from images tend to be powerful mathematics students. Further, we know that the ability to use images effectively in doing mathematics can be developed. When students are encouraged to develop mental images and use those images in mathematics, they show surprising growth. All students can learn to use images effectively. Thus, developing spatial sense should be a priority in school mathematics.

Quick Draw is an engaging mathematical activity that helps students develop their mental imagery. A figure such as the one shown below is presented briefly to students. They are asked to “Draw what you saw.” When students have drawn their figure, give them a second look. Finally, uncover the figure and ask students to describe what they saw. Encourage a wide range of interpretations. Some will see it as a two-dimensional figure while others may give it a three-dimensional interpretation. When they draw, they must work from a re-presented image since the figure shown is no longer in view. Finally, they are asked to describe what they saw and explain how they drew their sketch. As students listen to the ways others saw the figure, they are stimulated to reflect on their constructive activity and to consider other interpretations. It is often the case that students will describe new ways of viewing the figure as a direct result of listening to the descriptions of others.

The discussion of what they saw is a crucial component of the activity. Encourage students to talk about their drawings. Show enthusiasm for all interpretations. Be nonjudgmental, accepting all descriptions. Some students will be inspired by what others say. It is not unusual for five or more different ways of seeing the figure to be described. The whole class discussion of Quick Draw figures helps students get comfortable explaining their thinking to the class. There are no wrong answers. This carries over to lesson discussions of other topics. In learning mathematics, it is important that students become competent at articulating their thoughts as well as listening to other students.

Quick Draw activities

Two sample activities appear below. The first is the activity used in the video clip (see below). You can also download them in PDF format (requires Adobe Acrobat Reader) for use in your classroom. A second set of three Quick Draw activities is also avaialable here in PDF format. Additional activities are available in Grayson Wheatley’s book Quick Draw, available from Mathematics Learning. Other warm-up activities are explained in Coming to Know Number: A Mathematics Activity Resource for Elementary School Teachers by Grayson Wheatley and Anne M. Reynolds, also available from Mathematics Learning.

Quick Draw activity
Quick Draw activity

Watch the video

Quick Draw

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Watch Grayson Wheatley leading a class of 8th-graders through a Quick Draw exercise. (3:43)
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Quick Build

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Grayson Wheatley explains Quick Build, a 3D version of Quick Draw. (1:57)
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